Question

Prove that the following vectors are coplanar. The vectors are:
\[2\hat{i}-\hat{j}+\hat{k}\], \[\hat{i}-3\hat{j}-5\hat{k}\] and \[3\hat{i}-4\hat{j}-4\hat{k}\].

Answer 1

Hint: The given three vectors are coplanar if we can write one vector as the linear combination of other two vectors. So, we can prove that \[3\hat{i}-4\hat{j}-4\hat{k}=x(2\hat{i}-\hat{j}+\hat{k})+y(\hat{i}-3\hat{j}-5\hat{k})\], for some value of x and y, then we can say that the three given vectors are coplanar. The value of x and y can be any real number.

Complete step-by-step answer:
In the question, we have to prove that the vectors \[2\hat{i}-\hat{j}+\hat{k}\], \[\hat{i}-3\hat{j}-5\hat{k}\]and \[3\hat{i}-4\hat{j}-4\hat{k}\]. Are coplanar.
So here we will apply the concept that all the coplanar vectors are the linear combination of any one vector. So, in other words if we have three vectors \[\vec{a}\], \[\vec{b}\]and \[\vec{c}\]is said to be coplanar if we can write any vector let suppose \[\vec{c}=x\vec{a}+y\vec{b}\], which is the linear combination of other two vectors.
So applying this very concept, let the \[\vec{a}=2\hat{i}-\hat{j}+\hat{k}\], \[\vec{b}=\hat{i}-3\hat{j}-5\hat{k}\]and \[\vec{c}=3\hat{i}-4\hat{j}-4\hat{k}\]. So now we have to just show that:
 \[\begin{align}
  & \Rightarrow \vec{c}=x\vec{a}+y\vec{b} \\
 & \Rightarrow 3\hat{i}-4\hat{j}-4\hat{k}=x\left( 2\hat{i}-\hat{j}+\hat{k} \right)+y\left( \hat{i}-3\hat{j}-5\hat{k} \right) \\
\end{align}\]
Now, here if we have the value of x=1 and y=1, we get as follows:
\[\begin{align}
  & \Rightarrow 3\hat{i}-4\hat{j}-4\hat{k}=(1)\left( 2\hat{i}-\hat{j}+\hat{k} \right)+(1)\left( \hat{i}-3\hat{j}-5\hat{k} \right) \\
 & \Rightarrow 3\hat{i}-4\hat{j}-4\hat{k}=3\hat{i}-4\hat{j}-4\hat{k} \\
\end{align}\]
So we will get the vector \[3\hat{i}-4\hat{j}-4\hat{k}\] which is the sum of two vectors \[2\hat{i}-\hat{j}+\hat{k}\] and\[\hat{i}-3\hat{j}-5\hat{k}\].
Now, here we can easily see that the vector \[\vec{c}=x\vec{a}+y\vec{b}\], which is the linear combination of the vectors. Hence, the three vectors are coplanar.

Note: It can be noted that not only three vectors can be coplanar but we can have n number of vectors that can be coplanar if any one vector is the linear combination of all the vectors. Also, when we are adding two or more vectors, then we add the like terms to get the resultant vector.